Find the Taylor series about the given point for each of the functions. In each case find the radius of convergence. $f(x)=\sin x^{2}$ about $x_{0}=0$

Find the critical value(s) and rejection region(s) for the type of chi-square test with sample size n and level of significance $\alpha$. Two-tailed test, $n=31, \alpha=0.05$

Fill in the details of the proof that the symmetric groups $S_{\Delta}$ and $S_{\Omega}$ are isomorphic if $|\Delta|=|\Omega|$ as follows: let $\theta: \Delta \rightarrow \Omega$ be a bijection. Define

$$\varphi: S_{\Delta} \rightarrow S_{\Omega} \quad \text { by } \quad \varphi(\sigma)=\theta \circ \sigma \circ \theta^{-1} \quad \text { for all } \sigma \in S_{\Delta}$$

and prove the following: (a) $\varphi$ is well defined, that is, if $\sigma$ is a permutation of $\Delta$ then $\theta \circ \sigma \circ \theta^{-1}$ is a permutation of $\Omega$. (b) $\varphi$ is a bijection from $S_{\Delta}$ onto $S_{\Omega}$. [Find a 2-sided inverse for $\varphi$. (c) $\varphi$ is a homomorphism, that is,

$$\varphi(\sigma \circ \tau)=\varphi(\sigma) \circ \varphi(\tau)$$

How many Boolean functions are there from $Z^n _2$ into $Z_2$?

Take this test as you would take a test in class. Researchers surveyed 19,183 U.S. physicians, asking for the information below. location (region of the U.S.) income (dollars) employment status (private practice or an employee) benefits received (health insurance. liability coverage, etc.) specialty (cardiology, family medicine, radiology. etc.) time spent seeing patients per week (hours) (a) Identify the population and the sample. (h) Is the data collected qualitative, quantitative, or both? Explain your reasoning. (c) Determine the level of measurement for each item above. (d) Determine whether the study is an observational study or an experiment. Explain.

Find the relative maxima and relative minima, if any, of each function. $f(x)=x^{5 / 3}$

You are given the mass, damping, and spring constants of an undriven spring-mass system $m y^{\prime \prime}+\mu y^{\prime}+k y=0$. You are also given initial conditions. Use a numerical solver to (i) provide separate plots of the position versus time (y vs. t) and the velocity versus time (v vs. t) (ii) provide a combined plot of both position and velocity versus time, and (iji) provide a plot of the velocity versus position (v vs. y) in the yv phase plane. Choose a viewing window that highlights the important features of the solutions. m=1 kg, $\mu=0 \mathrm{kg} / \mathrm{s}$, $k=4 \mathrm{kg} / \mathrm{s}^{2}$, y(0)=-2 m, y'(0)=-2 m/s

The sine integral function is defined as $\mathrm{Si}(x)=\int_{0}^{x} \frac{\sin t}{t} d t$, where the integrand is defined to be 1 at x = 0. Express the solution of the initial-value problem $x^{3} \frac{d y}{d x}+2 x^{2} y=10 \sin x, \quad y(1)=0$ in terms of Si(x).

Let G = (V, E) be a simple undirected graph. A vertex cover of G is a subset $V^{\prime}$ of V such that for each edge $(v, w) \in E$, either $v \in V^{\prime}$ or $w \in V^{\prime}$. The size of a vertex cover $V^{\prime}$ is the number of vertices in $V^{\prime}$. An optimal vertex cover is a vertex cover of minimum size. An edge disjoint set for G is a subset $E^{\prime}$ of E such that for every pair of distinct edges $e_{1}=\left(v_{1}, w_{1}\right)$ and $e_2 = \left(v_{2}, w_{2}\right)$ in $E^{\prime}$ we have $\left\{v_{1}, w_{1}\right\} \cap\left\{v_{2}, w_{2}\right\}=\varnothing$. Show that the size of an optimal vertex cover of the complete graph on n vertices is n − 1.

Let G be the group of rigid motions in $\mathbb{R}^{3}$ of an octahedron. Show that |G| = 24.

Which of the period 2 homonuclear diatomic molecules are predicted to be paramagnetic?

If U is a universal set and S = P(U), the power set of U, then $\left(S, \cup, \cap,^{-}, \varnothing, U\right)$ is a Boolean algebra. State the bound and absorption laws for this Boolean algebra.

Under which of the following sets of conditions does a real gas behave most like an ideal gas, and for which conditions is a real gas expected to deviate from ideal behavior? Explain. (a) high pressure, small volume (b) high temperature, low pressure (c) low temperature, high pressure

The following problem explore the properties of matrix inverses. Suppose that A, B, and C are invertible matrices of the same size. Show that the product ABC is invertible and that $(\mathbf{A B C})^{-1}=\mathbf{C}^{-1} \mathbf{B}^{-1} \mathbf{A}^{-1}$.